Evaluating compositions of the form
f (
f−1 (
y )) and
f−1 (
f (
x ))
For any trigonometric function,
for all
in the proper domain for the given function. This follows from the definition of the inverse and from the fact that the range of
was defined to be identical to the domain of
However, we have to be a little more careful with expressions of the form
Compositions of a trigonometric function and its inverse
Is it correct that
No. This equation is correct if
belongs to the restricted domain
but sine is defined for all real input values, and for
outside the restricted interval, the equation is not correct because its inverse always returns a value in
The situation is similar for cosine and tangent and their inverses. For example,
Given an expression of the form f
−1 (f(θ)) where
evaluate.
If
is in the restricted domain of
If not, then find an angle
within the restricted domain of
such that
Then
Evaluating compositions of the form
f−1 (
g (
x ))
Now that we can compose a trigonometric function with its inverse, we can explore how to evaluate a composition of a trigonometric function and the inverse of another trigonometric function. We will begin with compositions of the form
For special values of
we can exactly evaluate the inner function and then the outer, inverse function. However, we can find a more general approach by considering the relation between the two acute angles of a right triangle where one is
making the other
Consider the sine and cosine of each angle of the right triangle in
[link] .
Because
we have
if
If
is not in this domain, then we need to find another angle that has the same cosine as
and does belong to the restricted domain; we then subtract this angle from
Similarly,
so
if
These are just the function-cofunction relationships presented in another way.
Given functions of the form
and
evaluate them.
If
then
If
then find another angle
such that
If
then
If
then find another angle
such that
Questions & Answers
Ayele, K., 2003. Introductory Economics, 3rd ed., Addis Ababa.
what's the difference between a firm and an industry
Abdul
firm is the unit which transform inputs to output where as industry contain combination of firms with similar production 😅😅
Abdulraufu
Suppose the demand function that a firm faces shifted from
Qd 120 3P
to
Qd 90 3P
and the supply function has shifted from
QS
20 2P
to
QS
10 2P .
a) Find the effect of this change on price and quantity.
b) Which of the changes in demand and supply is higher?
Demand curve shows that how supply and others conditions affect on demand of a particular thing and what percent demand increase whith increase of supply of goods
Israr
Hi Sir please how do u calculate Cross elastic demand and income elastic demand?
Abari
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